A novel lattice Boltzmann model for the Poisson equation

In this paper, a novel lattice Boltzmann model is proposed to solve the Poisson equation through modifying equilibrium distribution function. Compared with previous models, which can be viewed as the solvers to diffusion equation, the present model is a genuine solver to the Poisson equation, and the transient term derived by previous models is eliminated. Numerical solutions agree well with analytical solutions, which indicates the potential of the present model for solving the Poisson equation.     

Homogenization of Richardsʼ equation of van Genuchten–Mualem model

This paper is devoted to the homogenization of Richards? equation of van Genuchten–Mualem model, which is a nonlinear degenerate parabolic differential equation. It is usually used to model the motion of saturated–unsaturated water flow in porous media. We firstly apply the Kirchhoff transformation to the equation and obtain a simpler equivalent equation with a linear oscillated diffusion term. Then under the real assumption for van Genuchten–Mualem model, we obtain the homogenized equation based on the two-scale convergence theory. Some results on the first order corrector are also presented.     

On The Parametric Model of Western Boundary Outflow

A simple model equation for western boundary outflow in the Stommel model of the large scale ocean circulation is obtained by evaluating the potential vorticity equation at the western boundary. A series solution to this model equation demonstrates similar behavior to the boundary layer solution of the potential vorticity equation, in particular that “resonances” are present at a discrete series of parameter values which necessitate the addition of logarithms to the series; these resonances occur because the model equation has a logarithmic branch point at these values.     

一类非线性偏微分方程的四阶格子Boltzmann模型

通过Chapman-Enskog展开技术和多尺度分析,建立了一种新的D1Q4带修正项的四阶格子Boltzmann模型,一类非线性偏微分方程从连续的Boltzmann方程得到正确恢复．统一了KdV和Burgers等已知方程类型的格子BGK模型,还首次给出了组合KdV-Burgers,广义Burgers—Huxley等方程...     

Lattice Boltzmann model for two-dimensional generalized sine-Gordon equation

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical model based on lattice Boltmann method to obtain the numerical solutions of two-dimensional generalized sine-Gordon equation, including damped and undamped sine-Gordon equation. By choosing properly the conservation condition between the macroscopic quantity $u_t$ and the distribution functions and applying the Chapman-Enskog expansion, the governing equation is recovered correctly from the lattice Boltzmann equation. Moreover, the local equilibrium distribution function is obtained. The numerical results of the first three examples agree well with the analytic solutions, which indicates the lattice Boltzmann model is satisfactory and efficient. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given. Numerical experiments also show that the present scheme has a good long-time numerical behavior for the generalized sine-Gordon equation. Moreover, the model can also be applied to other two-dimensional nonlinear wave equations, such as nonlinear hyperbolic telegraph equation and Klein-Gordon equation.     

A model for a finite memory transport in the Fisher equation

A model for a finite memory effect in the Fisher equation had been presented by Cattaneo [Acad. Sci. 247 (1958) 431]. By this model the type of the governing equation is transformed from a parabolic type to a hyperbolic one. But the Cattaneo’s equation does not reduce to the logistic equation in the homogeneous regime. A new model is presented which conserves the parabolic generic equation as well as the reduction property. Memory effects are visualized in the two models through numerical computations of solutions.     

粉末注射成形填充过程的数值模拟

本文将粉末注射成形喂料在薄壁模腔中的流动视为二维流动，以流变学的基本方程为基础，建立了从动量方程、连续方程和热传递方程得到的描述PIM喂料充模二维流动的数学模型。在无滑移边界的条件下，推导了喂料熔体流导率的计算公式和压力场的控制方程，得到的压力场控制方程是一非线性椭圆偏微分方程．从而可用Galerkin方法进行数值求解，使模型的数值求解成为可能，为进一步对粉末注射成形进行计算机模拟和数值分析奠定了数学基础。     

Existence of Classical Solutions to a Stationary Simplified Quantum Energy-Transport Model in 1-Dimensional Space

The existence of classical solutions to a stationary simplified quantum energytransport model for semiconductor devices in 1-dimensional space is proved.The model consists of a nonlinear elliptic third-order equation for the electron density,including a temperature derivative,an elliptic nonlinear heat equation for the electron temperature,and the Poisson equation for the electric potential.The proof is based on an exponential variable transformation and the Leray-Schauder fixed-point theorem.     

一种考虑孔洞形状变化的损伤细观模型及其在孔洞闭合过程中的应用

本文在一种特殊的坐标系下,建立了非线性的基体材料,有限大的椭球体中含椭球形孔洞的损伤细观模型,考虑了孔洞形状的影响.得出的粘性约束方程(或称屈服面方程)除应力∑_ij,孔隙度f,幂硬化指数m外,还与孔洞的形状有关.通过曲线拟合的方法,对Gurson方程进行了修正,使之适合于非线性的基体材料、变形状孔洞的情形.最后将此模型用于分析非线性材料内部孔洞的闭合过程.     

Convergence d'un modèle à vitesses discrètes pour l'équation de Boltzmann-BGK

We consider a conservative and entropie discrete-velocity model for the Bathnagar-Gross-Krook (BGK) equation. In this model, the approximation of the Maxwellian is based on a discrete entropy minimization principle. First, we prove a consistency result for this approximation. Then, we demonstrate that the discrete-velocity model possesses a unique solution. Finally, the model is written in a continuous equation form, and we prove the convergence of its solution toward a solution of the BGK equation.     

Some integral equations related to random Gaussian processes

To calculate the Laplace transform of the integral of the square of a random Gaussian process, we consider a nonlinear Volterra-type integral equation. This equation is a Ward identity for the generating correlation function. It turns out that for an important class of correlation functions, this identity reduces to a linear ordinary differential equation. We present sufficient conditions for this equation to be integrable (the equation coefficients are constant). We calculate the Laplace transform exactly for some concrete random Gaussian processes such as the “Brownian bridge” model and the Ornstein-Uhlenbeck model.     

Viscosity solutions to a Cauchy problem of a phase-field model for solid-solid phase transitions

We investigate a partial differential equation which models solid-solid phase transitions. This model is for martensitic phase transitions driven by configurational force and its counterpart is for interface motion by mean curvature. Mathematically, this equation is a second-order nonlinear degenerate parabolic equation. And in multidimensional case, its principal part cannot be written into divergence form . We prove the existence and uniqueness of viscosity solution to a Cauchy problem for this model.     

一维半导体流体动力学模型的局部有界解

本文讨论了能量方程是压力一密度关系的一维半导体流体动力学模型方程,通过把欧拉－泊松方程变成拟线性波动方程,利用拟线性波动方程的局部解存在性,得到一维半导体流体动力学模型的局部解,并且解是有界的。     

Coloured noise for dispersion of contaminants in shallow waters

In this article, we explore the application of a set of stochastic differential equations called particle model in simulating the advection and diffusion of pollutants in shallow waters. The Fokker–Planck equation associated with this set of stochastic differential equations is interpreted as an advection–diffusion equation. This enables us to derive an underlying particle model that is exactly consistent with the advection–diffusion equation. Still, neither the advection–diffusion equation nor the related traditional particle model accurately takes into account the short-term spreading behaviour of particles. To improve the behaviour of the model shortly after the deployment of contaminants, a particle model forced by a coloured noise process is developed in this article. The use of coloured noise as a driving force unlike Brownian motion, enables to us to take into account the short-term correlated turbulent fluid flow velocity of the particles. Furthermore, it is shown that for long-term simulations of the dispersion of particles, both the particle due to Brownian motion and the particle model due to coloured noise are consistent with the advection–diffusion equation.     

Shallow water waves over an uneven bottom and an inhomogeneous KP equation

Three-dimensional shallow water waves over an uneven bottom are considered. The depth is assumed to be slow in variation. As a model, an inhomogeneous Kadomtsev-Petviashvili equation is presented. Some reductions of this equation are used to describe deformation of a line soliton due to the depth change. The model equation is valid for a wide class of two-dimensional nonlinear waves in inhomogeneous systems.     

Structured populations with distributed recruitment: from PDE to delay formulation

In this work, first, we consider a physiologically structured population model with a distributed recruitment process. That is, our model allows newly recruited individuals to enter the population at all possible individual states, in principle. The model can be naturally formulated as a first‐order partial integro‐differential equation, and it has been studied extensively. In particular, it is well posed on the biologically relevant state space of Lebesgue integrable functions. We also formulate a delayed integral equation (renewal equation) for the distributed birth rate of the population. We aim to illustrate the connection between the partial integro‐differential and the delayed integral equation formulation of the model utilising a recent spectral theoretic result. In particular, we consider the equivalence of the steady state problems in the two different formulations, which then lead us to characterise irreducibility of the semigroup governing the linear partial integro‐differential equation. Furthermore, using the method of characteristics, we investigate the connection between the time‐dependent problems. In particular, we prove that any (non‐negative) solution of the delayed integral equation determines a (non‐negative) solution of the partial differential equation and vice versa. The results obtained for the particular distributed states at birth model then lead us to present some very general results, which establish the equivalence between a general class of partial differential and delay equation, modelling physiologically structured populations. Copyright © 2016 John Wiley & Sons, Ltd.     

Properties of simple model problems for reacting gas flows

The compressible Navier–Stokes equations for reacting gases are extremely complex. Simpler models have been considered, and for these completely non-physical propagation speeds have been observed. These model problems are stiff, meaning that several different scales are present in the solution. Numerical solution of non-reacting flows almost always involves addition of extra dissipation. It will be shown that this action will render a totally wrong propagation speed for a simple model equation of reacting flows. This problem will be accentuated by increasing stiffness of the problem. Existence and uniqueness of a solution to this model equation is proved. The dependence of the propagation speed on the viscosity and a term governing the stiffness (comparable to the reaction rate for a more complete model) is investigated. A remedy for the wrong propagation speed for this simple model equation is proposed such that the speed is correct although the front is smeared out.     

Invariant approach to optimal investment–consumption problem: the constant elasticity of variance (CEV) model

The optimal investment–consumption problem under the constant elasticity of variance (CEV) model is solved using the invariant approach. Firstly, the invariance criteria for scalar linear second‐order parabolic partial differential equations in two independent variables are reviewed. The criteria is then employed to reduce the CEV model to one of the four Lie canonical forms. It is found that the invariance criteria help in transforming the original equation to the second Lie canonical form and with a proper parameter selection; the required transformation converts the original equation to the first Lie canonical form that is the heat equation. As a consequence, we find some new classes of closed‐form solutions of the CEV model for the case of reduction into heat equation and also into second Lie canonical form. The closed‐form analytical solution of the Cauchy initial value problems for the CEV model under investigation is also obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

A new boundary integral equation for molecular electrostatics with charges over whole space

In this paper, a new integral equation of electrostatics is proposed as an integral form of a basic dielectric continuum model, which is traditionally represented in a form of Poisson differential equation. As an application in protein simulations, the new integral equation is reduced to a second kind Fredholm boundary integral equation on the interface between the solute and solvent regions for a piecewise constant permittivity function, together with two new integral expressions for the electrostatics within the solute and solvent regions. The new integral equation and expressions work for any charge problem over the whole space (including the one with charges on the interface). This valuable feature is verified numerically for a dielectric sphere model with a point charge inside, outside, or on the sphere in this paper.